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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 34650.bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34650.bh1 | 34650z4 | \([1, -1, 0, -20701017, -36247127859]\) | \(86129359107301290313/9166294368\) | \(104409821785500000\) | \([2]\) | \(1966080\) | \(2.6933\) | |
34650.bh2 | 34650z2 | \([1, -1, 0, -1297017, -563171859]\) | \(21184262604460873/216872764416\) | \(2470316332176000000\) | \([2, 2]\) | \(983040\) | \(2.3468\) | |
34650.bh3 | 34650z3 | \([1, -1, 0, -325017, -1388399859]\) | \(-333345918055753/72923718045024\) | \(-830646725856601500000\) | \([2]\) | \(1966080\) | \(2.6933\) | |
34650.bh4 | 34650z1 | \([1, -1, 0, -145017, 7068141]\) | \(29609739866953/15259926528\) | \(173820100608000000\) | \([2]\) | \(491520\) | \(2.0002\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34650.bh have rank \(1\).
Complex multiplication
The elliptic curves in class 34650.bh do not have complex multiplication.Modular form 34650.2.a.bh
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.